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ATMS 535 Notes review

Thermodynamic Equations Review

 Cloud processes depend strongly on thermodynamic principles o changes impact latent heating o Static stability determines cloud lifting and vertical extent o Stability is influenced by water (vapor, phase changes) o Availability of moisture affects the rate at which cloud particles can grow by vapor deposition

Much of the seemingly complex thermodynamics in atmospheric sciences derives from several basic equations that are reviewed here:

IDEAL LAW

∗ [1] where p is , V is the of gas containing n moles, ∗ is the universal (8.314 J K-1 mol-1) and T is the . This law almost perfectly describes in the Earth’s atmosphere.

However, the form of [1] is not the most common form used in the atmospheric sciences. Usually, we are dealing with air, so we introduce a specific gas constant appropriate for dry air, -1 using ∗/ where is the molar of dry air (0.029 kg mol ), such that -1 -1 R=287.04 J K kg . Then, noting that = mass of air, and defining / from [1] we can rewrite [1] in its more common form:

[2]

If the composition of air were constant, then R would be a fixed number. However, moisture can -1 -1 vary in the atmosphere from 0 to 4% by mass, and since Rvapor = 461 J K kg , the value of R appropriate for a mixture of dry air and depends upon the vapor content. The conventional approach is to continue to use the gas constant for dry air, but to incorporate its moisture dependency into a new temperature measure called the Tv (see Lamb and Verlinde 1st edition, P221, or Wallace and Hobbs section 3.1.1).

HYDROSTATIC EQUATION [3]

Balance of vertical pressure gradient force with gravity. Combining with the law [2], we obtain this form:

[4] Whose consequence is the atmospheric scale height H = RT/g (~7.5 km).

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ATMS 535 Notes Thermodynamics review

FIRST LAW OF THERMODYNAMICS Expresses conservation of for a system in which external energy () input dq can change the u and/or do dw,

[5]

For the atmosphere, [5] is rarely used directly. Instead, we normalize by the mass of air, substitute , where is the (volume per mass, i.e. ), substitute

, where is the at constant volume, to give

[6]

With the product rule trick, , and noting that from the (R being the gas constant for air) we can substitute into [1] to obtain

[7]

Combining [7] with the hydrostatic equation [3], we arrive at a useful form of the first law:

[8]

This form is useful for instantly understanding how temperature changes with height under adiabatic lifting (dq=0), i.e. Γ, which defines the dry adiabatic -1 Γ (10 K km ). If one defines potential temperature as

[9] i.e. a variable with units of temperature whose changes are defined to be zero under adiabatic

changes, then one can integrate [7] with 0, so that =0. Integrating this from a reference pressure p0 where we define , to pressure p, we obtain the Poisson equation for the potential temperature:

/ [10]

CLAUSIUS‐CLAPEYRON EQUATION We will spend a little more later in the class discussing the physical basis for the Clausius- Clapeyron equation (see notes on Equilibria and Phase). Here, we will simply present it. The equation describes how the equilibrium (saturation) of water (es) changes with temperature T. The most common form of it in the atmospheric sciences is:

[11]

where is the of condensation and is the gas constant for water vapor. 2

ATMS 535 Notes Thermodynamics review

Note: Although [11] is technically an approximation it differs from the true expression by only 1 part in 1240 (ratio of the of liquid water to the molar volume of air).

Integration of [11] gives the quasi- es 40 exponential curve shown in the figure [hPa] (right): 30 Approx. 20 Exponential increase

10 6% per K

0 0 10 20 30 T [C]

The saturation vapor pressure represents the maximum vapor pressure of water before air becomes saturated. One consequence of the strong temperature dependence of es is that modest amounts of lifting (adiabatic cooling) easily lead to saturation given most typical moisture levels.

Relative humidity (RH) is defined as the ratio of the actual water vapor pressure to the saturation vapor pressure, i.e. RH = e/es.

Water vapor mixing ratio (usually given the symbol q or qv) is a common way to represent the water vapor content of the air, and is the mass of water vapor per mass of dry air. The vapor pressure e can be converted to a mixing ratio using:

[12]

Where p is the air pressure and (=0.622) is the ratio of molar of water Mw and dry air Ma. An approximate form of equation [12] is often used because e is small compared with p:

[13]

Saturation water vapor mixing ratio is calculated by applying [12] to the saturation vapor pressure es:

[14]

The homework – Thermodynamics and Cloud Vertical Structure – explores some of the consequences for cloud formation and structure of the presented here.

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